Statistics
Class 10 me humne data ka "center" (mean, median, mode) seekha. Lekin sirf average kaafi nahi — ye janna bhi zaroori hai ki data kitna faila (spread) hua hai. Is chapter me measures of dispersion padhenge: range, mean deviation, variance, aur standard deviation.
1. Dispersion kya hai
Do data sets ka same mean ho sakta hai, par ek tightly clustered ho aur doosra bikhra hua. Dispersion isi "bikhraav" ko naapta hai.
2. Range
\(\text{Range} = \text{Maximum value} - \text{Minimum value}.\)
Sabse simple, par sirf do extreme values par based — beech ka data ignore ho jaata hai.
3. Mean Deviation
Har value ka chosen center (mean ya median) se absolute antar nikaal kar unka average lena = mean deviation.
Mean \(\bar{x}\) ke baare me (ungrouped data, \(n\) values):
\[ \text{M.D.}(\bar{x}) = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|. \]
Frequency data ke liye (\(N = \sum f_i\)):
\[ \text{M.D.}(\bar{x}) = \frac{1}{N} \sum f_i\,|x_i - \bar{x}|. \]
4. Variance aur Standard Deviation
Mean deviation me absolute value handle karna mushkil hota hai, isliye antar ka square lete hain — yahi variance hai.
Variance \(\sigma^{2}\) = squared deviations ka average. Standard deviation \(\sigma = \sqrt{\text{variance}}\).
Ungrouped data:
\[ \sigma^{2} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^{2}, \qquad \sigma = \sqrt{\frac{1}{n} \sum (x_i - \bar{x})^{2}}. \]
Frequency data:
\[ \sigma^{2} = \frac{1}{N} \sum f_i (x_i - \bar{x})^{2}. \]
Data: \(2, 4, 6\). Mean \(\bar{x} = 4\). \[ \sigma^{2} = \frac{(2-4)^{2} + (4-4)^{2} + (6-4)^{2}}{3} = \frac{4 + 0 + 4}{3} = \frac{8}{3}, \quad \sigma = \sqrt{\tfrac{8}{3}} \approx 1.63. \]
5. Coefficient of Variation (CV)
Do alag data sets ki variability compare karne ke liye (jab unke means alag ho), CV use karte hain:
\[ \text{CV} = \frac{\sigma}{\bar{x}} \times 100\%. \]
Jiska CV zyada, wo data zyada variable (kam consistent); jiska CV kam, wo zyada consistent/stable.
Key Takeaways
- Dispersion = data ka spread; central value akela kaafi nahi.
- Range \(=\) max \(-\) min (simple par limited).
- Variance \(\sigma^{2} = \frac{1}{n}\sum(x_i - \bar{x})^{2}\); SD \(\sigma = \sqrt{\sigma^{2}}\).
- CV \(= \frac{\sigma}{\bar{x}} \times 100\%\) — consistency compare karne ke liye (kam CV = zyada consistent).